Heterogeneous gradient flows in the topology of fibered optimal transport
Jan Peszek, David Poyato
TL;DR
The paper introduces a fibered optimal transport topology that enforces fiber-preserving transport by penalizing cross-fiber mass exchange, yielding the fibered Wasserstein distance $W_{2,\nu}$ on the space $\mathcal{P}_{2,\nu}(\mathbb{R}^{2d})$. It develops a comprehensive gradient-flow theory in this fibered setting, proving a Metatheorem (Theorem A) for existence, uniqueness, and equivalence of gradient-flow notions under λ-convexity along generalized geodesics. The framework is then applied to first-order models with heterogeneous interactions (additive and multiplicative) and to singular second-order alignment models, notably multidimensional Kuramoto-type dynamics, establishing well-posedness, contraction, and mean-field limits (Theorem B). The results provide a rigorous variational structure for heterogeneous gradient flows and yield new insights into long-time behavior and collective dynamics with heterogeneity. Overall, the fibered calculus offers a robust toolset for analyzing PDEs with heterogeneous interactions and their mean-field limits in high dimensions.
Abstract
We introduce an optimal transport topology on the space of probability measures over a fiber bundle, which penalizes the transport cost from one fiber to another. For simplicity, we illustrate our construction in the Euclidean case $\mathbb{R}^d\times \mathbb{R}^d$, where we penalize the quadratic cost in the second component. Optimal transport becomes then constrained to happen along fixed fibers. Despite the degeneracy of the infinitely-valued and discontinuous cost, we prove that the space of probability measures $(\mathcal{P}_{2,ν}(\mathbb{R}^{2d}),W_{2,ν})$ with fixed marginal $ν\in \mathcal{P}(\mathbb{R}^d)$ in the second component becomes a Polish space under the fibered transport distance, which enjoys a weak Riemannian structure reminiscent of the one proposed by F. Otto for the classical quadratic Wasserstein space. Three fundamental issues are addressed: 1) We develop an abstract theory of gradient flows with respect to the new topology; 2) We show applications that identify a novel fibered gradient flow structure on a large class of evolution PDEs with heterogeneities; 3) We exploit our method to derive long-time behavior and global-in-time mean-field limits in a multidimensional Cucker-Smale-type alignment model with weakly singular coupling.